1 00:00:00,060 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:09,465 continue to offer high quality educational resources for free. 5 00:00:09,465 --> 00:00:12,600 To make a donation, or to view additional materials 6 00:00:12,600 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,150 --> 00:00:21,270 at ocw.mit.edu. 8 00:00:21,270 --> 00:00:26,900 PROFESSOR: Now, today I need to get started by finishing up 9 00:00:26,900 --> 00:00:28,480 what I did last time. 10 00:00:28,480 --> 00:00:31,560 Namely, talking about numerical methods. 11 00:00:31,560 --> 00:00:34,680 And I want to just carry out one example. 12 00:00:34,680 --> 00:00:40,660 And then I want to fill in one loose end. 13 00:00:40,660 --> 00:00:48,870 And then we'll talk about the unit overall. 14 00:00:48,870 --> 00:01:01,720 We were talking, last time, about numerical integration. 15 00:01:01,720 --> 00:01:05,580 I'm going to illustrate this just with the simplest example 16 00:01:05,580 --> 00:01:07,960 that I can. 17 00:01:07,960 --> 00:01:14,070 We're going to look at the integral from 1 to 2 of dx / x. 18 00:01:14,070 --> 00:01:16,720 Which we know perfectly well already 19 00:01:16,720 --> 00:01:21,310 is the log of x evaluated between 1 and 2, which is ln 2 20 00:01:21,310 --> 00:01:23,250 - ln 1. 21 00:01:23,250 --> 00:01:26,080 Which is just ln 2. 22 00:01:26,080 --> 00:01:29,750 Now, if you punch that into your calculator, 23 00:01:29,750 --> 00:01:32,610 you're going to get something like this. 24 00:01:32,610 --> 00:01:41,030 I hope I saved it here. 25 00:01:41,030 --> 00:01:41,530 Yeah. 26 00:01:41,530 --> 00:01:53,250 It's about 0.693147. 27 00:01:53,250 --> 00:01:56,150 That's more digits than we're going to get in our discussion 28 00:01:56,150 --> 00:01:56,650 here. 29 00:01:56,650 --> 00:02:02,490 Anyway, that's about how big this number is. 30 00:02:02,490 --> 00:02:06,390 And the numerical integration methods 31 00:02:06,390 --> 00:02:10,760 will give you about as much accuracy as you 32 00:02:10,760 --> 00:02:13,270 can get on the function itself. 33 00:02:13,270 --> 00:02:15,100 And, of course, some functions we 34 00:02:15,100 --> 00:02:17,040 may have more trouble approximating. 35 00:02:17,040 --> 00:02:19,400 But the function 1 / x, we know pretty well how to do, 36 00:02:19,400 --> 00:02:21,440 because we know how to divide. 37 00:02:21,440 --> 00:02:26,410 So since the function that we're integrating here is 1 / x, 38 00:02:26,410 --> 00:02:28,874 it's going to be not too difficult to get 39 00:02:28,874 --> 00:02:29,540 some arithmetic. 40 00:02:29,540 --> 00:02:31,740 Nevertheless, I'm going to do this 41 00:02:31,740 --> 00:02:34,210 in the simplest possible case. 42 00:02:34,210 --> 00:02:44,040 Namely, just with two intervals. 43 00:02:44,040 --> 00:02:47,020 Now, you really can't expect things to work so well 44 00:02:47,020 --> 00:02:48,040 with two intervals. 45 00:02:48,040 --> 00:02:52,560 That's a pretty ridiculous approximation to your function. 46 00:02:52,560 --> 00:02:54,830 When you have two intervals, that 47 00:02:54,830 --> 00:02:58,490 means you're looking at the graph of this hyperbola. 48 00:02:58,490 --> 00:03:04,510 And you have 1 here, and you have 2 here and you have 3/2. 49 00:03:04,510 --> 00:03:07,910 And you're really only keeping track of the values 50 00:03:07,910 --> 00:03:10,880 at these three spots. 51 00:03:10,880 --> 00:03:14,630 So the idea that you can approximate the area just 52 00:03:14,630 --> 00:03:16,940 by knowing the values of three places 53 00:03:16,940 --> 00:03:21,120 is a little bit of a stretch of the imagination. 54 00:03:21,120 --> 00:03:24,850 But we're going to try it anyway. 55 00:03:24,850 --> 00:03:33,320 Now, the trapezoidal rule is the following formula. 56 00:03:33,320 --> 00:03:38,980 It's delta x (1/2 the first value + the second value 57 00:03:38,980 --> 00:03:42,290 + 1/2 the third value). 58 00:03:42,290 --> 00:03:47,930 In this case, the pattern is 1/2, 1, 1, 1, 1, 1, 1/2. 59 00:03:47,930 --> 00:03:53,540 And in this case, delta x = 1/2 because this interval's 60 00:03:53,540 --> 00:03:55,120 of length 1. 61 00:03:55,120 --> 00:03:57,680 The b - a, right. 62 00:03:57,680 --> 00:03:59,990 Let's just point that out here. 63 00:03:59,990 --> 00:04:05,410 Here, b = 2. a = 1. b - a = 1. 64 00:04:05,410 --> 00:04:08,500 And the number n is 2. 65 00:04:08,500 --> 00:04:16,160 And so, delta x, which is (b - a) / n, is 1/2. 66 00:04:16,160 --> 00:04:17,580 So here's what we get. 67 00:04:17,580 --> 00:04:19,850 And let's just see what this number is. 68 00:04:19,850 --> 00:04:23,540 It's 1/2 of the value at here. 69 00:04:23,540 --> 00:04:25,540 Well, so let's just check what these values are. 70 00:04:25,540 --> 00:04:29,560 This value is 1, this value over here is 2/3, 71 00:04:29,560 --> 00:04:33,320 and the last value is 1/2. 72 00:04:33,320 --> 00:04:36,766 Because the function, of course, was y = 1 / x. 73 00:04:36,766 --> 00:04:38,640 And those were the three values that we have. 74 00:04:38,640 --> 00:04:42,030 So y_0, this one is y_0, this one is y_1, 75 00:04:42,030 --> 00:04:47,480 and this one is y_2. 76 00:04:47,480 --> 00:05:00,780 Now, here we have 1/2* 1 + 2/3 + 1/2 * 1/2. 77 00:05:00,780 --> 00:05:03,900 Now, on an exam, I don't expect you 78 00:05:03,900 --> 00:05:07,930 to add up long messes of numbers like this. 79 00:05:07,930 --> 00:05:09,700 When you have two numbers, I expect 80 00:05:09,700 --> 00:05:12,670 you to add them up if they're reasonable, or subtract them. 81 00:05:12,670 --> 00:05:14,530 Just as we do when we take antiderivatives. 82 00:05:14,530 --> 00:05:16,660 Like, for example, I don't want you 83 00:05:16,660 --> 00:05:18,860 to leave the answer to an integration like this 84 00:05:18,860 --> 00:05:19,490 in this form. 85 00:05:19,490 --> 00:05:21,514 I want you to simplify it at least down to here. 86 00:05:21,514 --> 00:05:23,180 And I of course don't expect you to know 87 00:05:23,180 --> 00:05:24,880 the numerical approximation. 88 00:05:24,880 --> 00:05:27,417 But I certainly expect you to be able to do that. 89 00:05:27,417 --> 00:05:29,000 On the other hand, when the arithmetic 90 00:05:29,000 --> 00:05:32,360 gets a little bit long, you can relax a little bit. 91 00:05:32,360 --> 00:05:35,210 But I did carry this out on my calculator. 92 00:05:35,210 --> 00:05:41,240 Unless I'm mistaken, it's about 0.96. 93 00:05:41,240 --> 00:05:44,820 It's pretty far off. 94 00:05:44,820 --> 00:05:47,484 So remember what it was. 95 00:05:47,484 --> 00:05:49,650 It's what you get when you get these straight lines. 96 00:05:49,650 --> 00:05:52,170 And there are these little extra pieces of junk there. 97 00:05:52,170 --> 00:05:55,870 Now, don't trust that too much, but the point 98 00:05:55,870 --> 00:06:00,540 is that it's far off. 99 00:06:00,540 --> 00:06:09,760 So now, let's take a look at Simpson's Rule. 100 00:06:09,760 --> 00:06:13,500 And I claim that Simpson's Rule is surprisingly accurate. 101 00:06:13,500 --> 00:06:15,500 In this case, really, even a little more than it 102 00:06:15,500 --> 00:06:17,410 deserves to be. 103 00:06:17,410 --> 00:06:29,400 The formula is (delta x / 3) (y_0 + 4 y_1 + y_2). 104 00:06:29,400 --> 00:06:32,460 So the pattern is 1, 4, 1, or 1, 4 105 00:06:32,460 --> 00:06:39,200 and then it alternates 2's and 4's until 4, 1 at the very end. 106 00:06:39,200 --> 00:06:43,060 And if I just plug in the numbers now, what I get 107 00:06:43,060 --> 00:06:47,340 is 1/6, because delta x = 1/2 again. 108 00:06:47,340 --> 00:06:50,600 And the value for y_0 was 1. 109 00:06:50,600 --> 00:06:55,120 And the value for y_1 was 2/3. 110 00:06:55,120 --> 00:06:59,350 And the value for y_2 was 1/2. 111 00:06:59,350 --> 00:07:03,740 So here's the estimate in this case. 112 00:07:03,740 --> 00:07:08,540 And this one I did carry out carefully. 113 00:07:08,540 --> 00:07:16,880 And it came out to 0.69444. 114 00:07:16,880 --> 00:07:18,810 Which is actually pretty impressive, 115 00:07:18,810 --> 00:07:20,490 if you think about it. 116 00:07:20,490 --> 00:07:26,050 Given what the logarithm is. 117 00:07:26,050 --> 00:07:31,740 Now, what's going on with Simpson's Rule in general 118 00:07:31,740 --> 00:07:33,240 is this. 119 00:07:33,240 --> 00:07:43,360 If you-- Simpson's minus the exact answer, 120 00:07:43,360 --> 00:07:46,970 in absolute value, is approximately of the size 121 00:07:46,970 --> 00:07:50,850 of (delta x)^4. 122 00:07:50,850 --> 00:07:52,730 That's really the way it behaves. 123 00:07:52,730 --> 00:08:03,300 Which means that if delta x is about 1/10, 124 00:08:03,300 --> 00:08:05,670 so if we had divided this up into 10 intervals, 125 00:08:05,670 --> 00:08:08,270 which we didn't, but if we'd divided it up into 10 126 00:08:08,270 --> 00:08:11,585 intervals, then you could expect that delta x-- 127 00:08:11,585 --> 00:08:15,320 the error would be about 10^(-4). 128 00:08:15,320 --> 00:08:21,090 In other words, four digits of accuracy here for this thing. 129 00:08:21,090 --> 00:08:25,940 But the exact analysis of this, a more careful analysis 130 00:08:25,940 --> 00:08:27,640 of this, is in your textbook. 131 00:08:27,640 --> 00:08:30,820 And I'm not going to do it. 132 00:08:30,820 --> 00:08:33,750 But I just want to point out that it is an effective method. 133 00:08:33,750 --> 00:08:35,830 It really does give you nice four-digit 134 00:08:35,830 --> 00:08:38,510 with manageable-- you could even really do it by hand. 135 00:08:38,510 --> 00:08:42,040 It's so convenient, the Simpson's Rule. 136 00:08:42,040 --> 00:08:46,140 Whereas the other rules aren't really that impressive 137 00:08:46,140 --> 00:08:51,790 as far as giving fairly accurate answers. 138 00:08:51,790 --> 00:08:54,330 The last little remark to make is 139 00:08:54,330 --> 00:08:57,520 that the reason is that Simpson's Rule is 140 00:08:57,520 --> 00:08:59,300 matching a parabola. 141 00:08:59,300 --> 00:09:03,650 And somehow the parabola follows this curve better. 142 00:09:03,650 --> 00:09:05,890 It's giving the exact answer. 143 00:09:05,890 --> 00:09:07,570 So I'll mention this. 144 00:09:07,570 --> 00:09:23,590 Simpson's Rule is derived using the exact answer 145 00:09:23,590 --> 00:09:32,770 for all degree 2 polynomials. 146 00:09:32,770 --> 00:09:36,680 In other words, parabolas. 147 00:09:36,680 --> 00:09:38,410 All parabolas. 148 00:09:38,410 --> 00:09:40,450 But even all the ones of lower degree. 149 00:09:40,450 --> 00:09:43,360 So straight lines would work, and constants 150 00:09:43,360 --> 00:09:44,390 would work as well. 151 00:09:44,390 --> 00:09:47,770 Whereas the other ones only work for, say, straight lines. 152 00:09:47,770 --> 00:09:51,200 The trapezoidal rule only works for straight lines. 153 00:09:51,200 --> 00:09:53,110 But there is a weird accident. 154 00:09:53,110 --> 00:09:55,650 It turns out that it also works for cubics. 155 00:09:55,650 --> 00:09:59,460 Once you get the formulas, it works for cubics. 156 00:09:59,460 --> 00:10:06,630 So it's also exact for cubics. 157 00:10:06,630 --> 00:10:12,400 And that's what explains the fourth order validity. 158 00:10:12,400 --> 00:10:14,900 The last thing that I want to point out 159 00:10:14,900 --> 00:10:17,820 is that this is extremely vague, what I said there. 160 00:10:17,820 --> 00:10:20,340 And you should be a little bit cautious about it. 161 00:10:20,340 --> 00:10:30,890 You need to watch out for 1/x for x near 0. 162 00:10:30,890 --> 00:10:34,120 All bets are off if the function is singular. 163 00:10:34,120 --> 00:10:36,250 And there's a lot of area under there. 164 00:10:36,250 --> 00:10:40,740 And it's also true that if the derivative messes up, 165 00:10:40,740 --> 00:10:41,770 you're in trouble too. 166 00:10:41,770 --> 00:10:44,480 You really need for the function to be nice and smooth in order 167 00:10:44,480 --> 00:10:46,700 for Simpson's Rule to work. 168 00:10:46,700 --> 00:10:48,740 This is wath out. 169 00:10:48,740 --> 00:10:55,920 That's a real wath out, but we'll try to-- Watch out. 170 00:10:55,920 --> 00:10:58,290 Watch out for whenever x near 0. 171 00:10:58,290 --> 00:11:00,770 Then this thing doesn't work. 172 00:11:00,770 --> 00:11:03,710 This thing really depends on bounds on derivatives. 173 00:11:03,710 --> 00:11:05,780 But I'm going to be relatively vague about that. 174 00:11:05,780 --> 00:11:12,780 I'm not attempting to give you an error analysis here. 175 00:11:12,780 --> 00:11:16,520 OK, so if you were doing this on an exam, 176 00:11:16,520 --> 00:11:21,600 how do you remember this strange pattern of numbers? 177 00:11:21,600 --> 00:11:27,250 The one thing that I want to recommend to you is, as a way 178 00:11:27,250 --> 00:11:35,620 of remembering it, so the one mnemonic device, 179 00:11:35,620 --> 00:11:39,340 we'll call it a mnemonic device here, 180 00:11:39,340 --> 00:11:42,850 for remembering what it is that you're doing, 181 00:11:42,850 --> 00:11:48,990 is to remind yourself of what happens for the simplest 182 00:11:48,990 --> 00:11:50,060 possible case. 183 00:11:50,060 --> 00:11:53,730 Which is f(x) = 1. 184 00:11:53,730 --> 00:11:57,260 It seems very modest, but if it doesn't give you 185 00:11:57,260 --> 00:11:59,410 the exact answer for f(x) = 1, you've 186 00:11:59,410 --> 00:12:02,220 got the wrong weightings. 187 00:12:02,220 --> 00:12:06,120 And here, if you check out what happens in the first formula 188 00:12:06,120 --> 00:12:14,110 here, y_0 / 2 + y_1 +..., well, we'll go all the way to y_(n-1) 189 00:12:14,110 --> 00:12:14,720 + y_n / 2. 190 00:12:17,770 --> 00:12:19,450 If you check that formula out here, 191 00:12:19,450 --> 00:12:21,440 this is the trapezoidal rule. 192 00:12:21,440 --> 00:12:25,220 If you check it out for this case, then what you get 193 00:12:25,220 --> 00:12:29,050 is that this is equal to delta x times what? 194 00:12:29,050 --> 00:12:30,629 Well, all of these are 1's. 195 00:12:30,629 --> 00:12:32,170 And how many are there in the middle? 196 00:12:32,170 --> 00:12:34,440 There are n - 1 of them in the middle. 197 00:12:34,440 --> 00:12:38,740 So it's 1/2 + n - 1 + 1/2. 198 00:12:38,740 --> 00:12:40,030 At the tail end. 199 00:12:40,030 --> 00:12:43,680 So all told it's delta x n. 200 00:12:43,680 --> 00:12:50,980 And I remind you that delta x = b - a / n. 201 00:12:50,980 --> 00:12:57,030 So, delta x, this thing, is equal to b - a. 202 00:12:57,030 --> 00:12:58,930 And that's just as it should be. 203 00:12:58,930 --> 00:13:03,130 What we just calculated is an approximation to this integral 204 00:13:03,130 --> 00:13:04,080 here. 205 00:13:04,080 --> 00:13:07,660 Which is just the area of the rectangle of base b - 206 00:13:07,660 --> 00:13:09,210 a and height 1. 207 00:13:09,210 --> 00:13:12,080 Which of course is b - a. 208 00:13:12,080 --> 00:13:15,540 So this is the check that you got your weighted average 209 00:13:15,540 --> 00:13:18,610 correct here. 210 00:13:18,610 --> 00:13:20,880 You've put the correct weightings on everything. 211 00:13:20,880 --> 00:13:23,400 And you can do this same thing with Simpson's Rule. 212 00:13:23,400 --> 00:13:31,030 And match up those quantities. 213 00:13:31,030 --> 00:13:36,390 There was a question in the room at some point. 214 00:13:36,390 --> 00:13:38,440 No, OK. 215 00:13:38,440 --> 00:13:42,680 So now, the next thing I want to do for you 216 00:13:42,680 --> 00:13:49,730 is the loose end which I left hanging. 217 00:13:49,730 --> 00:13:55,280 Namely, I want to compute that mysterious constant square root 218 00:13:55,280 --> 00:13:56,680 of pi / 2. 219 00:13:56,680 --> 00:14:00,260 This is really one of the most famous computations 220 00:14:00,260 --> 00:14:02,420 in calculus. 221 00:14:02,420 --> 00:14:04,740 And it's a very, very clever trick. 222 00:14:04,740 --> 00:14:09,160 I certainly don't expect you to come up with this trick. 223 00:14:09,160 --> 00:14:11,360 I certainly wouldn't have myself. 224 00:14:11,360 --> 00:14:15,750 But it's an important thing to calculate. 225 00:14:15,750 --> 00:14:17,560 And it's just very useful. 226 00:14:17,560 --> 00:14:19,310 So I'm going to tell you about it. 227 00:14:19,310 --> 00:14:22,400 And it's just on the subject that we're dealing with 228 00:14:22,400 --> 00:14:28,260 in this unit; namely, slicing. 229 00:14:28,260 --> 00:14:30,400 Or adding up. 230 00:14:30,400 --> 00:14:38,850 So the first step, which is just something that we already did, 231 00:14:38,850 --> 00:14:50,800 was that we found the volume under this curve. 232 00:14:50,800 --> 00:14:54,180 This bell-shaped curve, e^(-r^2). 233 00:14:54,180 --> 00:15:00,300 But rotated around an axis. 234 00:15:00,300 --> 00:15:09,020 Rotated around this axis. 235 00:15:09,020 --> 00:15:10,680 Around this way. 236 00:15:10,680 --> 00:15:12,360 So we figured that out. 237 00:15:12,360 --> 00:15:15,280 And that was a relatively short computation. 238 00:15:15,280 --> 00:15:18,870 I'm just going to remind you, it goes by shells. 239 00:15:18,870 --> 00:15:23,500 We integrate the whole range from 0 to infinity. 240 00:15:23,500 --> 00:15:28,530 And we have 2 pi r 2 pi r e^(-r^2) dr. 241 00:15:28,530 --> 00:15:30,532 So this again is the circumference of the shell. 242 00:15:30,532 --> 00:15:32,240 This is the height of the shell, and this 243 00:15:32,240 --> 00:15:34,650 is the thickness of the shell. 244 00:15:34,650 --> 00:15:45,280 Circumference, height, thickness. 245 00:15:45,280 --> 00:15:47,790 So we're just taking a little piece here and sweeping it 246 00:15:47,790 --> 00:15:48,520 around. 247 00:15:48,520 --> 00:15:52,000 And then adding up. 248 00:15:52,000 --> 00:15:58,250 And then this antiderivative is pi-- -pi e^(-r^2), 249 00:15:58,250 --> 00:16:00,420 evaluated at 0 and infinity. 250 00:16:00,420 --> 00:16:03,140 And we worked this out last time. 251 00:16:03,140 --> 00:16:06,520 This is pi. 252 00:16:06,520 --> 00:16:12,700 It's pi (1 - 0). 253 00:16:12,700 --> 00:16:13,200 Which is pi. 254 00:16:13,200 --> 00:16:15,700 So the conclusion is that V = pi. 255 00:16:15,700 --> 00:16:20,220 We already know that. 256 00:16:20,220 --> 00:16:22,770 Now, the problem that we want to deal with now 257 00:16:22,770 --> 00:16:31,460 is the problem not of a volume, but an area. 258 00:16:31,460 --> 00:16:32,990 And this looks quite different. 259 00:16:32,990 --> 00:16:37,340 And of course the answer is going to be different. 260 00:16:37,340 --> 00:16:38,250 But let's do it. 261 00:16:38,250 --> 00:16:40,610 So this is this question mark here. 262 00:16:40,610 --> 00:16:47,970 And I'm going to do the one from minus infinity to infinity. 263 00:16:47,970 --> 00:16:50,000 And I'll relate it to what we talked 264 00:16:50,000 --> 00:16:54,210 about earlier in this unit, in just a couple of minutes 265 00:16:54,210 --> 00:17:00,520 when I show you the procedure that we're going to follow. 266 00:17:00,520 --> 00:17:02,710 So here's the quantity and now, what 267 00:17:02,710 --> 00:17:07,210 this is interpreted as is the area under this bell curve. 268 00:17:07,210 --> 00:17:22,950 This time, Q is really an area. 269 00:17:22,950 --> 00:17:26,370 Now, what's going to turn out to happen, is this. 270 00:17:26,370 --> 00:17:27,920 This is the trick. 271 00:17:27,920 --> 00:17:31,480 We're going to compute V in a different way. 272 00:17:31,480 --> 00:17:35,920 And you'll see it laid out in just a second. 273 00:17:35,920 --> 00:17:42,870 We will compute V by slices. 274 00:17:42,870 --> 00:17:45,370 We're going to slice it like a piece of bread here. 275 00:17:45,370 --> 00:17:48,640 We're going to solve for that same thing here. 276 00:17:48,640 --> 00:17:52,120 And then, amazingly, what's going to happen is that we will 277 00:17:52,120 --> 00:17:57,720 discover that V = Q^2. 278 00:17:57,720 --> 00:18:03,170 That's going to be what's going to come out. 279 00:18:03,170 --> 00:18:07,009 And that's the end of the computation that we want. 280 00:18:07,009 --> 00:18:08,800 Because actually we already know what V is. 281 00:18:08,800 --> 00:18:11,290 We don't want to read this equation forward. 282 00:18:11,290 --> 00:18:12,890 We want to read it the other way. 283 00:18:12,890 --> 00:18:18,310 We want to say Q^2 = V, which we already know is pi. 284 00:18:18,310 --> 00:18:29,770 And so Q is equal to the square root of pi. 285 00:18:29,770 --> 00:18:33,680 I haven't shown this yet, this is the weird part. 286 00:18:33,680 --> 00:18:35,560 And I'm going to put it in a little box 287 00:18:35,560 --> 00:18:38,220 so that we know that this is what we need to check. 288 00:18:38,220 --> 00:18:43,070 We need to check this fact here. 289 00:18:43,070 --> 00:18:45,000 We haven't done that yet. 290 00:18:45,000 --> 00:18:51,500 Now, let me connect this with what we did a few days ago. 291 00:18:51,500 --> 00:18:54,690 With what I called one of the important functions 292 00:18:54,690 --> 00:19:02,540 of mathematics besides the ones you already know. 293 00:19:02,540 --> 00:19:04,810 And so the function that we were faced with, 294 00:19:04,810 --> 00:19:15,080 and that we discussed, was this one. 295 00:19:15,080 --> 00:19:19,380 And then, we were interested in the value at infinity. 296 00:19:19,380 --> 00:19:23,780 We were interested in this. 297 00:19:23,780 --> 00:19:26,930 Which, if you draw a picture of it, 298 00:19:26,930 --> 00:19:29,650 and you draw the same bell curve, 299 00:19:29,650 --> 00:19:32,610 that's the area under half. of it. 300 00:19:32,610 --> 00:19:35,850 That's the area starting from 0 and going to infinity. 301 00:19:35,850 --> 00:19:37,470 That's the area under half. 302 00:19:37,470 --> 00:19:46,270 So this chunk is F of infinity. 303 00:19:46,270 --> 00:19:50,000 And now I hope that this part of the connection 304 00:19:50,000 --> 00:19:52,140 is not meant to be fancy. 305 00:19:52,140 --> 00:19:58,760 The idea here is that Q = 2 F(infinity). 306 00:19:58,760 --> 00:20:00,190 This number here. 307 00:20:00,190 --> 00:20:04,380 And so F F(infinity) is equal to the square root of pi 308 00:20:04,380 --> 00:20:10,900 over 2, if we believe what we said on the last panel. 309 00:20:10,900 --> 00:20:14,490 And that was the thing that I drew a picture of on the board. 310 00:20:14,490 --> 00:20:18,390 Namely, the graph of F looked like this. 311 00:20:18,390 --> 00:20:20,670 And there was this asymptote, which 312 00:20:20,670 --> 00:20:25,020 was the limit F(x) tends to square root of pi 313 00:20:25,020 --> 00:20:27,650 over 2, as x goes to infinity. 314 00:20:27,650 --> 00:20:30,560 That was that limiting value. 315 00:20:30,560 --> 00:20:33,910 Which is F of infinity. 316 00:20:33,910 --> 00:20:40,460 So this is the asymptote. 317 00:20:40,460 --> 00:20:43,350 And now I've explained the connection 318 00:20:43,350 --> 00:20:47,260 between what we claimed before, which was quite mysterious, 319 00:20:47,260 --> 00:20:50,840 and what we're actually going to be able to check now. 320 00:20:50,840 --> 00:21:07,170 Concretely, by making this computation. 321 00:21:07,170 --> 00:21:09,980 So how in the world can you get something like this. 322 00:21:09,980 --> 00:21:14,490 What's in that orange box there, that V = Q^2. 323 00:21:14,490 --> 00:21:19,300 Again, the technique is to use slices here. 324 00:21:19,300 --> 00:21:21,860 And I'm going to have to draw you a 3-D picture 325 00:21:21,860 --> 00:21:24,970 to visualize the slice. 326 00:21:24,970 --> 00:21:27,360 Let's do that. 327 00:21:27,360 --> 00:21:30,520 I'm going to draw three axes now, 328 00:21:30,520 --> 00:21:33,640 because we're now going to be in three-dimensional space, 329 00:21:33,640 --> 00:21:37,250 and I want you to imagine the x-axis as coming out 330 00:21:37,250 --> 00:21:40,320 of the blackboard, the y-axis is horizontal, 331 00:21:40,320 --> 00:21:41,980 and there's a new axis, which I'll call 332 00:21:41,980 --> 00:21:45,090 the z-axis, which is going up. 333 00:21:45,090 --> 00:21:46,860 So what's happening here is that I'm 334 00:21:46,860 --> 00:21:50,810 thinking of this-- This is, if you like, some kind of side 335 00:21:50,810 --> 00:21:51,320 view. 336 00:21:51,320 --> 00:21:53,653 And this is a view where I've tilted things a little bit 337 00:21:53,653 --> 00:21:57,080 up to the top. 338 00:21:57,080 --> 00:21:59,974 Now, the distribution, or you could 339 00:21:59,974 --> 00:22:01,640 think of this target in the plane, where 340 00:22:01,640 --> 00:22:04,110 the most likely places to hit were in the middle 341 00:22:04,110 --> 00:22:05,510 and it died off. 342 00:22:05,510 --> 00:22:07,390 As we went down. 343 00:22:07,390 --> 00:22:11,400 Now, I want to draw a picture of this graph. 344 00:22:11,400 --> 00:22:15,240 I'm going to draw a picture of e^(-r^2). 345 00:22:15,240 --> 00:22:18,090 And it's basically a hump. 346 00:22:18,090 --> 00:22:22,420 So I'm going to take the first-- the slice along y = 0. 347 00:22:22,420 --> 00:22:23,880 The y = 0 slice. 348 00:22:23,880 --> 00:22:26,300 And I claim that that goes up like this. 349 00:22:26,300 --> 00:22:28,310 And then comes back down. 350 00:22:28,310 --> 00:22:32,100 Let me shade this in, so that you can see 351 00:22:32,100 --> 00:22:34,140 what kind of a slice this is. 352 00:22:34,140 --> 00:22:38,679 This is supposed to be along this vertical plane here. 353 00:22:38,679 --> 00:22:40,220 Which is coming out of the blackboard 354 00:22:40,220 --> 00:22:42,409 and coming towards you. 355 00:22:42,409 --> 00:22:43,200 And that's a slice. 356 00:22:43,200 --> 00:22:46,630 Now, I'm going to draw one more slice so that you 357 00:22:46,630 --> 00:22:49,000 can see what's happening. 358 00:22:49,000 --> 00:22:52,310 I'm going to draw a slice at another place. 359 00:22:52,310 --> 00:22:52,860 Along here. 360 00:22:52,860 --> 00:22:55,030 This will be y = b. 361 00:22:55,030 --> 00:22:56,709 Some other level. 362 00:22:56,709 --> 00:22:58,500 And now I'm going to show you what happens. 363 00:22:58,500 --> 00:23:02,960 What happens is that the hump dies down a little bit. 364 00:23:02,960 --> 00:23:06,750 So the bump is just a little bit lower. 365 00:23:06,750 --> 00:23:10,284 And it's going to look a little bit the same way. 366 00:23:10,284 --> 00:23:11,950 But it's just going to be a bit smaller. 367 00:23:11,950 --> 00:23:15,380 So there's another slice here. 368 00:23:15,380 --> 00:23:16,940 Like that. 369 00:23:16,940 --> 00:23:20,890 And I want to give a name to these slices. 370 00:23:20,890 --> 00:23:25,090 I'm going to call this A(b). 371 00:23:25,090 --> 00:23:36,130 That is, the area of the b slice. 372 00:23:36,130 --> 00:23:39,790 Under the surface. 373 00:23:39,790 --> 00:23:41,010 OK? 374 00:23:41,010 --> 00:23:41,870 Yes, question. 375 00:23:41,870 --> 00:23:47,256 STUDENT: [INAUDIBLE] 376 00:23:47,256 --> 00:23:48,380 PROFESSOR: Yeah, the solid. 377 00:23:48,380 --> 00:23:48,890 Yeah. 378 00:23:48,890 --> 00:23:52,830 We're trying to figure out this volume here, 379 00:23:52,830 --> 00:23:56,780 which is the one we started out with, by slices. 380 00:23:56,780 --> 00:23:59,500 So first I have to think of-- I'm going to visualize-- 381 00:23:59,500 --> 00:24:01,970 So here I didn't even visualize. 382 00:24:01,970 --> 00:24:03,490 I took a cross section and I thought 383 00:24:03,490 --> 00:24:06,580 about how to spin it around without actually doing that 384 00:24:06,580 --> 00:24:07,954 in three-dimensional space. 385 00:24:07,954 --> 00:24:10,120 But now I'm going to take a different kind of slice. 386 00:24:10,120 --> 00:24:12,260 I'm going to take that same bump, which 387 00:24:12,260 --> 00:24:13,680 is a three-dimensional object. 388 00:24:13,680 --> 00:24:16,050 I'm going to lay it down on a plane. 389 00:24:16,050 --> 00:24:17,540 Which looks like this. 390 00:24:17,540 --> 00:24:20,850 And then it's a bump here. 391 00:24:20,850 --> 00:24:22,260 It's a hump. 392 00:24:22,260 --> 00:24:27,350 And now I'm going to try to slice it by various planes. 393 00:24:27,350 --> 00:24:29,370 STUDENT: [INAUDIBLE] 394 00:24:29,370 --> 00:24:34,270 PROFESSOR: So one way of defining 395 00:24:34,270 --> 00:24:37,609 the bump, as you just suggested, is you take this curve 396 00:24:37,609 --> 00:24:39,150 and you rotate it around this z-axis. 397 00:24:39,150 --> 00:24:41,400 So in other words, you make this the axis of rotation, 398 00:24:41,400 --> 00:24:43,505 you spin it around. 399 00:24:43,505 --> 00:24:44,130 That's correct. 400 00:24:44,130 --> 00:24:48,020 So that shows you that the peaks as you go down here 401 00:24:48,020 --> 00:24:50,740 are going to descend the same way. 402 00:24:50,740 --> 00:24:54,140 But I don't want to draw those lines. 403 00:24:54,140 --> 00:24:57,780 I want to imagine what the parallel slices are. 404 00:24:57,780 --> 00:24:59,610 Because I don't want to get cross slices. 405 00:24:59,610 --> 00:25:01,950 I want all slices parallel to the same thing. 406 00:25:01,950 --> 00:25:04,997 STUDENT: [INAUDIBLE] 407 00:25:04,997 --> 00:25:05,580 PROFESSOR: OK. 408 00:25:05,580 --> 00:25:13,070 This is not particularly easy to visualize. 409 00:25:13,070 --> 00:25:17,960 Now, here's the formula for volume by slices. 410 00:25:17,960 --> 00:25:23,900 The formula for volume by slices is that you add up 411 00:25:23,900 --> 00:25:28,770 the areas of the slices. 412 00:25:28,770 --> 00:25:31,050 That's how you do it. 413 00:25:31,050 --> 00:25:32,465 You take each slice. 414 00:25:32,465 --> 00:25:33,840 You add the cross-sectional area, 415 00:25:33,840 --> 00:25:36,170 and then you take a little thickness, dy, 416 00:25:36,170 --> 00:25:38,330 and then you add all of them up. 417 00:25:38,330 --> 00:25:40,950 Because this is extending over the whole plane, 418 00:25:40,950 --> 00:25:43,630 we're going to have to go all the way from minus infinity 419 00:25:43,630 --> 00:25:45,100 to plus infinity. 420 00:25:45,100 --> 00:25:56,940 And this is the formula for volumes by slicing. 421 00:25:56,940 --> 00:26:00,300 And now our goal, in order to do this calculation, 422 00:26:00,300 --> 00:26:04,620 we're going to just fix y is equal to some b. 423 00:26:04,620 --> 00:26:06,840 We're just going to fix one of these slices. 424 00:26:06,840 --> 00:26:13,454 And we're going to calculate A(b). 425 00:26:13,454 --> 00:26:14,870 That's what we need to do in order 426 00:26:14,870 --> 00:26:23,030 to make this procedure succeed. 427 00:26:23,030 --> 00:26:27,740 This is the only place where this method works. 428 00:26:27,740 --> 00:26:32,120 But it's an important one. 429 00:26:32,120 --> 00:26:34,195 In order to make it work, I'm going 430 00:26:34,195 --> 00:26:37,760 to have to again draw the plot from a different point of view. 431 00:26:37,760 --> 00:26:43,550 I'm going to do the top view. 432 00:26:43,550 --> 00:26:48,750 So I want to look down on this x-y plane here. 433 00:26:48,750 --> 00:26:52,020 This is the x-direction, and here's the y-direction. 434 00:26:52,020 --> 00:26:56,930 And then again I want to draw my slice. 435 00:26:56,930 --> 00:27:00,760 My slice is here. 436 00:27:00,760 --> 00:27:04,860 At y = b. 437 00:27:04,860 --> 00:27:07,430 So we're just right on top of it. 438 00:27:07,430 --> 00:27:09,404 And it's coming up at some kind of bump. 439 00:27:09,404 --> 00:27:11,070 Here, with a little higher in the middle 440 00:27:11,070 --> 00:27:15,970 and going down on the sides. 441 00:27:15,970 --> 00:27:20,210 Now, the formula for the height is this. 442 00:27:20,210 --> 00:27:25,290 If I take a distance r here, the formula for the height 443 00:27:25,290 --> 00:27:27,290 of the bump is e^(-r^2). 444 00:27:31,200 --> 00:27:33,390 I'll store that over here. 445 00:27:33,390 --> 00:27:37,910 e^(-r^2) is the height at this place. 446 00:27:37,910 --> 00:27:41,020 If this distance to the origin is r. 447 00:27:41,020 --> 00:27:43,360 That's true all the way around. 448 00:27:43,360 --> 00:27:48,622 And in terms of b and x, we can figure out that 449 00:27:48,622 --> 00:27:49,580 by this right triangle. 450 00:27:49,580 --> 00:27:52,400 This height is b, and this distance is x. 451 00:27:52,400 --> 00:27:55,940 So r^2 = b^2 + x^2. 452 00:28:02,060 --> 00:28:02,560 Question. 453 00:28:02,560 --> 00:28:15,350 STUDENT: [INAUDIBLE] 454 00:28:15,350 --> 00:28:19,105 PROFESSOR: The question is, is that the x-y plane. 455 00:28:19,105 --> 00:28:23,280 So the answer is that over here I cleverly used the letter r. 456 00:28:23,280 --> 00:28:27,010 I avoided using y's and z's or anything. 457 00:28:27,010 --> 00:28:28,970 And over here, this is the distance r. 458 00:28:28,970 --> 00:28:31,650 And you like, this is z, going up. 459 00:28:31,650 --> 00:28:33,430 That's the way to think of it. 460 00:28:33,430 --> 00:28:36,180 So that all of the letters are consistent. 461 00:28:36,180 --> 00:28:39,170 So I just avoided giving it a name. 462 00:28:39,170 --> 00:28:41,960 That's good, that's exactly the point. 463 00:28:41,960 --> 00:28:43,160 Alright. 464 00:28:43,160 --> 00:28:46,870 So now, I claim I have a formula for r^2. 465 00:28:46,870 --> 00:28:48,397 And so I can write this down. 466 00:28:48,397 --> 00:28:49,230 This e^(-b^2 + x^2). 467 00:28:58,620 --> 00:29:01,934 But now I'm going to use the rule of exponents. 468 00:29:01,934 --> 00:29:04,350 Which is that this is the same as e^(-b^2) times e^(-x^2). 469 00:29:07,120 --> 00:29:11,330 And that's going to be the main way in which we 470 00:29:11,330 --> 00:29:15,590 use the particular function that we're dealing with here. 471 00:29:15,590 --> 00:29:20,940 That's really the main step, amazingly. 472 00:29:20,940 --> 00:29:33,470 So now I get to compute what A(b) is. 473 00:29:33,470 --> 00:29:37,960 A(b) is the area under a curve. 474 00:29:37,960 --> 00:29:40,300 So it's going to be, let me write it over here, 475 00:29:40,300 --> 00:29:47,160 A(b) is the area under this curve here. 476 00:29:47,160 --> 00:29:51,580 Which is some constant times-- so if you imagine, 477 00:29:51,580 --> 00:29:53,270 call this thing the name c. 478 00:29:53,270 --> 00:29:57,070 Under some curve, ce^(-x^2). 479 00:29:57,070 --> 00:29:59,040 Where the c is equal to e^(-b^2). 480 00:30:06,380 --> 00:30:07,560 That's what our slice is. 481 00:30:07,560 --> 00:30:09,060 In fact, it looks like one of those. 482 00:30:09,060 --> 00:30:12,970 It looks like one of those bumps. 483 00:30:12,970 --> 00:30:14,630 Here's its formula again. 484 00:30:14,630 --> 00:30:17,850 It's the integral from minus infinity to infinity 485 00:30:17,850 --> 00:30:19,500 of e^(-b^2) e^(-x^2) dx. 486 00:30:28,960 --> 00:30:31,560 We just recopied what I had up there. 487 00:30:31,560 --> 00:30:37,430 And this is the height at each value of x, with b fixed. 488 00:30:37,430 --> 00:30:42,630 And now, so we have a lot of steps here. 489 00:30:42,630 --> 00:30:44,680 But each of them is very elementary. 490 00:30:44,680 --> 00:30:47,380 The first one was just that law of exponents. 491 00:30:47,380 --> 00:30:49,520 That we could split the two into products. 492 00:30:49,520 --> 00:30:51,970 Now I'm going to make that splitting even further. 493 00:30:51,970 --> 00:30:53,420 This is a constant. 494 00:30:53,420 --> 00:30:55,150 It's not varying with x. 495 00:30:55,150 --> 00:30:59,060 So I'm going to factor it out of the integral. 496 00:30:59,060 --> 00:31:02,230 This is e^(-b^2) times the integral from minus infinity 497 00:31:02,230 --> 00:31:03,850 to infinity of e^(-x^2) dx. 498 00:31:06,390 --> 00:31:09,890 So this might look frightening, but actually it's just 499 00:31:09,890 --> 00:31:11,110 the property of an integral. 500 00:31:11,110 --> 00:31:13,180 All integrals have this kind of property. 501 00:31:13,180 --> 00:31:17,090 You can always factor out a constant. 502 00:31:17,090 --> 00:31:20,900 And now here comes the remarkable thing. 503 00:31:20,900 --> 00:31:26,920 This is e^(-b^2) times a number which is now familiar to us. 504 00:31:26,920 --> 00:31:27,980 What is this number? 505 00:31:27,980 --> 00:31:29,280 This is what we're looking for. 506 00:31:29,280 --> 00:31:40,550 This is our unknown, Q. So I've computed A(b), 507 00:31:40,550 --> 00:31:44,260 and now I'm ready to finish the problem off. 508 00:31:44,260 --> 00:31:48,720 A(b) = e^(-b^2) Q. Q is that strange number which we 509 00:31:48,720 --> 00:31:49,380 don't know yet. 510 00:31:49,380 --> 00:31:51,850 What it is. 511 00:31:51,850 --> 00:31:55,570 So now I'm going to compute the whole volume. 512 00:31:55,570 --> 00:31:57,850 The whole volume, remember, it's over there, 513 00:31:57,850 --> 00:32:02,540 it's minus infinity to infinity, A(y) dy. 514 00:32:02,540 --> 00:32:05,300 And now I'm just going to plug in the formula that we've found 515 00:32:05,300 --> 00:32:07,860 for A. Now I'm doing this for each b, 516 00:32:07,860 --> 00:32:09,740 so I'm doing it varying over all b's. 517 00:32:09,740 --> 00:32:12,710 So I have the integral from minus infinity to infinity. 518 00:32:12,710 --> 00:32:15,490 And here I have e^(-y^2). 519 00:32:15,490 --> 00:32:17,620 I've replaced b by y. 520 00:32:17,620 --> 00:32:21,640 And now I have Q. And I have dy. 521 00:32:21,640 --> 00:32:23,970 I just recopied what I had over there 522 00:32:23,970 --> 00:32:27,100 into the formula for slicing. 523 00:32:27,100 --> 00:32:29,920 And now, I'm going to do this trick of factoring out 524 00:32:29,920 --> 00:32:32,260 the constant a second time. 525 00:32:32,260 --> 00:32:33,860 This is a constant. 526 00:32:33,860 --> 00:32:35,180 It doesn't depend on y. 527 00:32:35,180 --> 00:32:38,030 It's the same for all y, it just will factor out. 528 00:32:38,030 --> 00:32:40,880 So this is the same as Q times the integral from minus 529 00:32:40,880 --> 00:32:47,350 infinity to infinity, e^(-y^2) dy. 530 00:32:47,350 --> 00:32:52,200 And now, lo and behold, this expression here. 531 00:32:52,200 --> 00:32:54,510 Of course, notice how I defined Q. Let's 532 00:32:54,510 --> 00:32:56,950 go back carefully to where Q is defined. 533 00:32:56,950 --> 00:33:02,900 Here's Q. This t is a dummy variable. 534 00:33:02,900 --> 00:33:04,370 It doesn't matter what I call it. 535 00:33:04,370 --> 00:33:05,790 I can call it x, I can call it u, 536 00:33:05,790 --> 00:33:07,620 I can call it v. In this case, I've 537 00:33:07,620 --> 00:33:10,020 given it two different names. 538 00:33:10,020 --> 00:33:13,030 At this stage, I called it x. 539 00:33:13,030 --> 00:33:14,860 And at this stage I'm calling it y. 540 00:33:14,860 --> 00:33:17,970 But it's the same variable. 541 00:33:17,970 --> 00:33:20,650 And so this little chunk is Q and altogether I have two 542 00:33:20,650 --> 00:33:23,320 of them, for Q^2 being the total. 543 00:33:23,320 --> 00:33:32,050 And that's the end of the argument. 544 00:33:32,050 --> 00:33:33,010 It's a real miracle. 545 00:33:33,010 --> 00:33:45,920 STUDENT: [INAUDIBLE] 546 00:33:45,920 --> 00:33:47,580 PROFESSOR: Great question. 547 00:33:47,580 --> 00:33:49,490 The question is, wait a minute. 548 00:33:49,490 --> 00:33:53,620 As y changes, doesn't x change. 549 00:33:53,620 --> 00:33:58,120 And so then this wouldn't be a constant. 550 00:33:58,120 --> 00:34:01,740 So that's the way in which we've used the letters x and y 551 00:34:01,740 --> 00:34:04,110 in this whole course. 552 00:34:04,110 --> 00:34:07,710 When you get to 18.02, you'll almost never do that. 553 00:34:07,710 --> 00:34:10,807 Always y and x will be different variables. 554 00:34:10,807 --> 00:34:12,640 And they won't have to depend on each other. 555 00:34:12,640 --> 00:34:15,700 Now, let me show you where on this picture the x and the y 556 00:34:15,700 --> 00:34:17,610 are. 557 00:34:17,610 --> 00:34:21,000 We've got a whole x-y plane, and here I'm fixing y = b, 558 00:34:21,000 --> 00:34:23,220 y isn't varying. 559 00:34:23,220 --> 00:34:25,340 Whereas x is changing. 560 00:34:25,340 --> 00:34:29,000 So, in other words, I don't have a relationship between x and y, 561 00:34:29,000 --> 00:34:29,900 unless I fix it. 562 00:34:29,900 --> 00:34:32,690 In this case I've decided that y is going to be constant. 563 00:34:32,690 --> 00:34:35,400 For all x. 564 00:34:35,400 --> 00:34:39,110 Over here, I made a computation. 565 00:34:39,110 --> 00:34:41,380 And I have a Q, which is just a single number. 566 00:34:41,380 --> 00:34:44,830 No matter which b I took, it didn't matter which. 567 00:34:44,830 --> 00:34:46,130 No matter which y equals b. 568 00:34:46,130 --> 00:34:47,930 Of course, I changed the name to b so it 569 00:34:47,930 --> 00:34:50,630 wouldn't be so jarring to you. 570 00:34:50,630 --> 00:34:53,920 But in fact this b was y all along. 571 00:34:53,920 --> 00:34:57,460 It's just that the x varied completely independently 572 00:34:57,460 --> 00:34:57,960 of the y. 573 00:34:57,960 --> 00:35:00,300 I could fix the y and vary the x, I could fix the x 574 00:35:00,300 --> 00:35:03,390 and vary the y. 575 00:35:03,390 --> 00:35:06,590 So it's a different use of the letters. 576 00:35:06,590 --> 00:35:07,720 From what you're used to. 577 00:35:07,720 --> 00:35:10,360 It happens that y is not a function of x. 578 00:35:10,360 --> 00:35:11,171 In this case. 579 00:35:11,171 --> 00:35:11,670 Yes. 580 00:35:11,670 --> 00:35:16,455 STUDENT: [INAUDIBLE] 581 00:35:16,455 --> 00:35:17,080 PROFESSOR: Yes. 582 00:35:17,080 --> 00:35:23,030 STUDENT: [INAUDIBLE] 583 00:35:23,030 --> 00:35:25,610 PROFESSOR: The question is, because I'm 584 00:35:25,610 --> 00:35:29,000 rotating around the z-axis, doesn't x change exactly 585 00:35:29,000 --> 00:35:30,210 as much as y does. 586 00:35:30,210 --> 00:35:33,790 What happens is that x and y are symmetric variables. 587 00:35:33,790 --> 00:35:36,870 They can be treated equally. 588 00:35:36,870 --> 00:35:41,660 But if I decide to take slices with respect to y being fixed 589 00:35:41,660 --> 00:35:45,070 and x varying, then of course they're now separated, 590 00:35:45,070 --> 00:35:47,410 and I have a separate role for the x and a separate role 591 00:35:47,410 --> 00:35:48,870 for the y. 592 00:35:48,870 --> 00:35:50,420 Or if I'd sliced it the other way, 593 00:35:50,420 --> 00:35:51,920 I would have gotten the same answer. 594 00:35:51,920 --> 00:35:54,560 I just would have reversed the roles of x and y. 595 00:35:54,560 --> 00:35:56,100 So what's happening is that x and y 596 00:35:56,100 --> 00:36:01,150 are on equal footing with each other in this picture, 597 00:36:01,150 --> 00:36:02,650 and I could've sliced the other way. 598 00:36:02,650 --> 00:36:04,610 I would have gotten the same answer. 599 00:36:04,610 --> 00:36:07,660 That's more or less the answer to your question. 600 00:36:07,660 --> 00:36:13,180 OK. 601 00:36:13,180 --> 00:36:16,300 Now I have given you a review sheet, 602 00:36:16,300 --> 00:36:19,150 and I want to run through, briefly, 603 00:36:19,150 --> 00:36:23,390 what's going to be on the exam. 604 00:36:23,390 --> 00:36:28,140 And this list of exam questions is 605 00:36:28,140 --> 00:36:30,120 what's going to be on the exam. 606 00:36:30,120 --> 00:36:33,390 There are, sorry this is not displayed correctly. 607 00:36:33,390 --> 00:36:35,110 So, exam questions, but now I'm just 608 00:36:35,110 --> 00:36:36,490 going to show you what they are. 609 00:36:36,490 --> 00:36:38,090 There are five questions on the exam. 610 00:36:38,090 --> 00:36:46,580 They are completely parallel to what you got last year. 611 00:36:46,580 --> 00:36:48,100 So you should look at that test. 612 00:36:48,100 --> 00:36:50,570 It's worth looking at. 613 00:36:50,570 --> 00:36:54,750 And you'll see in the descriptions on this sheet 614 00:36:54,750 --> 00:37:00,410 that what I'm describing is what's on that test. 615 00:37:00,410 --> 00:37:04,160 So what's going to happen is - and this is also 616 00:37:04,160 --> 00:37:06,790 posted on the Web - is that you'll 617 00:37:06,790 --> 00:37:11,020 be expected to calculate some definite integrals using 618 00:37:11,020 --> 00:37:13,750 the fundamental theorem of calculus. 619 00:37:13,750 --> 00:37:16,000 Do a numerical approximation. 620 00:37:16,000 --> 00:37:17,930 There'll be a Riemann, a trapezoidal rule 621 00:37:17,930 --> 00:37:19,550 and a Simpson's Rule. 622 00:37:19,550 --> 00:37:22,770 Calculate areas and volumes. 623 00:37:22,770 --> 00:37:25,510 And then some other cumulative sum. 624 00:37:25,510 --> 00:37:29,990 Either an average value or probability or perhaps work. 625 00:37:29,990 --> 00:37:32,410 And sketch a function which is given 626 00:37:32,410 --> 00:37:37,380 in this form as an integral. 627 00:37:37,380 --> 00:37:42,020 So those are the questions, and you'll 628 00:37:42,020 --> 00:37:46,080 see by the example of last year's exam exactly the style. 629 00:37:46,080 --> 00:37:48,540 They're really going to be very similar. 630 00:37:48,540 --> 00:37:49,450 Yes, question. 631 00:37:49,450 --> 00:38:01,582 STUDENT: [INAUDIBLE] 632 00:38:01,582 --> 00:38:02,790 PROFESSOR: OK, good question. 633 00:38:02,790 --> 00:38:05,560 So the question is, for Riemann sums, 634 00:38:05,560 --> 00:38:08,850 what's the difference between upper and lower, 635 00:38:08,850 --> 00:38:19,750 and right and left? 636 00:38:19,750 --> 00:38:25,300 So here we have a Riemann sum. 637 00:38:25,300 --> 00:38:29,720 And I'm going to give you a picture which is, 638 00:38:29,720 --> 00:38:33,045 maybe this function y = 1 / x, which was the one that we 639 00:38:33,045 --> 00:38:37,900 were discussing earlier. 640 00:38:37,900 --> 00:38:40,420 If you take the function y = 1 / x 641 00:38:40,420 --> 00:38:43,540 and you break it up into pieces here, 642 00:38:43,540 --> 00:38:45,370 however it doesn't matter how many pieces, 643 00:38:45,370 --> 00:38:48,390 let's just say there are four of them. 644 00:38:48,390 --> 00:38:53,020 Then the lower Riemann sum is the staircase 645 00:38:53,020 --> 00:38:55,490 which fits underneath. 646 00:38:55,490 --> 00:39:01,270 So this one is a picture of the lower sum. 647 00:39:01,270 --> 00:39:03,220 It's always less. 648 00:39:03,220 --> 00:39:08,960 And in the case of a decreasing function, it's going to be, 649 00:39:08,960 --> 00:39:19,530 so since if you like, since 1 / x decreases, 650 00:39:19,530 --> 00:39:27,710 the lower sum equals the right sum. 651 00:39:27,710 --> 00:39:29,980 You can see that visually on this picture. 652 00:39:29,980 --> 00:39:32,730 The values you're going to select 653 00:39:32,730 --> 00:39:38,360 are going to be the right ends of the rectangles. 654 00:39:38,360 --> 00:39:40,980 The upper sum is the left one. 655 00:39:40,980 --> 00:39:43,340 Now, if the function wiggles up and down, 656 00:39:43,340 --> 00:39:45,700 then you have to pick whichever side is appropriate. 657 00:39:45,700 --> 00:39:47,366 Or maybe it'll be a point in the middle, 658 00:39:47,366 --> 00:39:54,360 if the maximum is achieved in the middle. 659 00:39:54,360 --> 00:39:55,370 Yeah, another question. 660 00:39:55,370 --> 00:40:02,549 STUDENT: [INAUDIBLE] 661 00:40:02,549 --> 00:40:03,340 PROFESSOR: Correct. 662 00:40:03,340 --> 00:40:09,630 If the function is increasing, then the lower sum 663 00:40:09,630 --> 00:40:11,170 is the left sum. 664 00:40:11,170 --> 00:40:13,630 So it just exactly reverses what's here. 665 00:40:13,630 --> 00:40:17,920 So this is decreasing, lower sum is right-hand sum. 666 00:40:17,920 --> 00:40:24,400 Increasing, lower sum is left-hand sum. 667 00:40:24,400 --> 00:40:26,135 STUDENT: [INAUDIBLE] 668 00:40:26,135 --> 00:40:26,760 PROFESSOR: Yes. 669 00:40:26,760 --> 00:40:44,549 STUDENT: [INAUDIBLE] 670 00:40:44,549 --> 00:40:45,590 PROFESSOR: Good question. 671 00:40:45,590 --> 00:40:47,570 Suppose you're faced with a function like this 672 00:40:47,570 --> 00:40:48,660 in this last problem. 673 00:40:48,660 --> 00:40:52,270 Which, generally, these are the trickiest problems. 674 00:40:52,270 --> 00:40:54,150 And the question is, how are you ever going 675 00:40:54,150 --> 00:40:55,790 to be able to decide on an asymptote, 676 00:40:55,790 --> 00:40:58,110 even whether there is an asymptote. 677 00:40:58,110 --> 00:41:01,930 And the answer is, you're not. 678 00:41:01,930 --> 00:41:05,211 It's going to be pretty tricky to get 679 00:41:05,211 --> 00:41:07,460 keep track of what's happening as it goes to infinity. 680 00:41:07,460 --> 00:41:08,960 We had an example on the homework 681 00:41:08,960 --> 00:41:10,470 where is was oscillating and it's 682 00:41:10,470 --> 00:41:12,100 very unclear what's going on. 683 00:41:12,100 --> 00:41:15,100 You have to do a very long analysis for that. 684 00:41:15,100 --> 00:41:20,892 So in fact, just don't worry about that now. 685 00:41:20,892 --> 00:41:22,350 At the very end of the class, we'll 686 00:41:22,350 --> 00:41:24,180 talk a little bit about these asymptotes. 687 00:41:24,180 --> 00:41:27,620 And really, the first issue is whether they exist or not. 688 00:41:27,620 --> 00:41:30,275 And that's even something. 689 00:41:30,275 --> 00:41:32,150 That's a serious question which we'll address 690 00:41:32,150 --> 00:41:33,800 at the very end of this course. 691 00:41:33,800 --> 00:41:35,680 STUDENT: [INAUDIBLE] 692 00:41:35,680 --> 00:41:36,680 PROFESSOR: That's right. 693 00:41:36,680 --> 00:41:40,840 It's not going to be anything that complicated. 694 00:41:40,840 --> 00:41:42,350 Other questions? 695 00:41:42,350 --> 00:41:44,100 We we still have a five whole minutes, 696 00:41:44,100 --> 00:41:50,440 and I have an example to give, if nobody has a question. 697 00:41:50,440 --> 00:41:50,940 Yeah. 698 00:41:50,940 --> 00:41:51,773 STUDENT: [INAUDIBLE] 699 00:41:51,773 --> 00:41:55,710 PROFESSOR: The question, uh, will I tell you 700 00:41:55,710 --> 00:41:59,410 which one of what to use? 701 00:41:59,410 --> 00:42:00,910 STUDENT: [INAUDIBLE] PROFESSOR: When 702 00:42:00,910 --> 00:42:02,576 I tell you the numeric approximation is, 703 00:42:02,576 --> 00:42:04,450 you'll see on the exam. 704 00:42:04,450 --> 00:42:06,450 The practice exam that you have. 705 00:42:06,450 --> 00:42:09,220 I will ask you for all three. 706 00:42:09,220 --> 00:42:11,620 I will ask you for the Riemann sum, the trapezoidal rule, 707 00:42:11,620 --> 00:42:12,660 and the Simpson's rule. 708 00:42:12,660 --> 00:42:16,104 I'm guaranteeing you they'll all three be on the exam. 709 00:42:16,104 --> 00:42:18,020 I'm guaranteeing that every single thing which 710 00:42:18,020 --> 00:42:20,930 is on that piece of paper is on the exam. 711 00:42:20,930 --> 00:42:23,865 And you'll see it on the exam that you've got. 712 00:42:23,865 --> 00:42:27,420 It's exactly parallel to what's there. 713 00:42:27,420 --> 00:42:31,341 STUDENT: [INAUDIBLE] 714 00:42:31,341 --> 00:42:32,840 PROFESSOR: So with areas and volume, 715 00:42:32,840 --> 00:42:36,720 the question is will I tell you which method to use. 716 00:42:36,720 --> 00:42:48,340 So let's discuss that. 717 00:42:48,340 --> 00:42:58,180 So with areas and volumes, there's basically-- 718 00:42:58,180 --> 00:43:00,170 So this is always true with areas. 719 00:43:00,170 --> 00:43:07,040 And it's true with volumes of revolution. 720 00:43:07,040 --> 00:43:08,970 By the way you should read this sheet. 721 00:43:08,970 --> 00:43:12,740 Not everything that's on here have I said. 722 00:43:12,740 --> 00:43:15,020 But you should read it. 723 00:43:15,020 --> 00:43:16,470 Because it's all relevant. 724 00:43:16,470 --> 00:43:19,820 So with volumes of revolution, you always work your way 725 00:43:19,820 --> 00:43:25,600 back to some 2-D diagram. 726 00:43:25,600 --> 00:43:30,910 So there's some 2-D diagram which is always-- 727 00:43:30,910 --> 00:43:33,090 two-dimensional diagram, which is always 728 00:43:33,090 --> 00:43:34,960 connected with these problems. 729 00:43:34,960 --> 00:43:37,960 I mean, something this hard is really just too hard 730 00:43:37,960 --> 00:43:39,080 to do on an exam, right? 731 00:43:39,080 --> 00:43:41,538 I mean, I'm not going to ask you something this complicated 732 00:43:41,538 --> 00:43:42,472 on the exam. 733 00:43:42,472 --> 00:43:44,180 Because this involves a three-dimensional 734 00:43:44,180 --> 00:43:47,200 visualization. 735 00:43:47,200 --> 00:43:50,270 But once you're down to 2-D, you're 736 00:43:50,270 --> 00:43:52,180 supposed to be able to handle it. 737 00:43:52,180 --> 00:43:56,710 Now, what's the main issue after you've got your 2-D diagram? 738 00:43:56,710 --> 00:43:59,990 The main issue is, do you want to integrate with respect 739 00:43:59,990 --> 00:44:06,950 to dx or dy? 740 00:44:06,950 --> 00:44:13,850 And the answer is that it will depend. 741 00:44:13,850 --> 00:44:16,810 And if there's one that's going to cause 742 00:44:16,810 --> 00:44:21,310 you incredible difficulty, and I feel that you're not 743 00:44:21,310 --> 00:44:24,940 able to dodge it, then I might give you a hint 744 00:44:24,940 --> 00:44:27,750 and say you'd better use shells, or you'd better 745 00:44:27,750 --> 00:44:31,370 use disks or washers or something like that. 746 00:44:31,370 --> 00:44:34,760 But if I feel that you're grown up enough to figure out 747 00:44:34,760 --> 00:44:37,470 which one it is, because one of them is so ridiculous 748 00:44:37,470 --> 00:44:40,840 you say forget it, immediately, after thinking about it. 749 00:44:40,840 --> 00:44:43,460 Then I won't tell you which one. 750 00:44:43,460 --> 00:44:44,870 Because I figure, in other words, 751 00:44:44,870 --> 00:44:47,160 I don't want you to waste your time. 752 00:44:47,160 --> 00:44:50,580 But I'm willing to waste a minute or two of your time 753 00:44:50,580 --> 00:44:57,420 on a wild goose chase. 754 00:44:57,420 --> 00:45:01,380 So let me give you an example of this. 755 00:45:01,380 --> 00:45:14,670 Suppose you're looking at the curve y between 0 and x - x^3. 756 00:45:14,670 --> 00:45:17,380 So this is some kind of lump. 757 00:45:17,380 --> 00:45:18,460 Like this. 758 00:45:18,460 --> 00:45:25,080 It goes from 0 to 1, because the right-hand side is 0 at 0 and 1 759 00:45:25,080 --> 00:45:25,921 here. 760 00:45:25,921 --> 00:45:26,920 It's some kind of thing. 761 00:45:26,920 --> 00:45:29,850 And there are these two possibilities. 762 00:45:29,850 --> 00:45:33,220 One of them is to do shells. 763 00:45:33,220 --> 00:45:41,040 And then, so this is supposed to be rotated around the y-axis. 764 00:45:41,040 --> 00:45:45,940 In this case. 765 00:45:45,940 --> 00:45:48,519 And the same would apply, actually, to the area problem. 766 00:45:48,519 --> 00:45:50,560 So I'm doing a slightly more complicated problem. 767 00:45:50,560 --> 00:45:53,660 But you could ask for the area underneath this, and so forth. 768 00:45:53,660 --> 00:45:54,480 OK. 769 00:45:54,480 --> 00:45:57,400 So we can integrate this dx, or we can integrate this dy. 770 00:45:57,400 --> 00:46:00,130 This indicates that I'm deciding that this 771 00:46:00,130 --> 00:46:02,600 is going to be of thickness dx, and I'm integrating dx. 772 00:46:02,600 --> 00:46:04,940 So that's a choice that I'm making. 773 00:46:04,940 --> 00:46:07,160 Now, the minute I made that choice 774 00:46:07,160 --> 00:46:10,000 I know that these are shells. 775 00:46:10,000 --> 00:46:12,980 Because they sweep around this way and that makes them shells. 776 00:46:12,980 --> 00:46:15,460 Cylindrical shells. 777 00:46:15,460 --> 00:46:19,210 And if I do that, the setup is this. 778 00:46:19,210 --> 00:46:25,540 It's 2 2 pi x (x - x^3) dx. 779 00:46:25,540 --> 00:46:28,340 Now, I claim that when you get to this point, 780 00:46:28,340 --> 00:46:30,580 you already know you've won. 781 00:46:30,580 --> 00:46:32,920 Because this is an easy integral to calculate. 782 00:46:32,920 --> 00:46:34,640 So you're done here. 783 00:46:34,640 --> 00:46:37,840 You're happy. 784 00:46:37,840 --> 00:46:42,930 Now, if you happened to say, oh gee, I hate to do this. 785 00:46:42,930 --> 00:46:47,040 I want to do something clever, you 786 00:46:47,040 --> 00:46:54,220 could try to do it with cutting this way. 787 00:46:54,220 --> 00:46:56,500 Let's do this. 788 00:46:56,500 --> 00:47:01,610 And this would be the dy thickness. 789 00:47:01,610 --> 00:47:04,760 And then when you sweep this around, 790 00:47:04,760 --> 00:47:10,560 you get what we call a washer. 791 00:47:10,560 --> 00:47:14,710 Which is really just the difference of two disks. 792 00:47:14,710 --> 00:47:24,560 So the shape here is this thing swung around this axis. 793 00:47:24,560 --> 00:47:26,110 And it looks like this. 794 00:47:26,110 --> 00:47:29,270 So it's going to be the difference of radii. 795 00:47:29,270 --> 00:47:32,580 So what's the formula for this? 796 00:47:32,580 --> 00:47:37,070 It's some integral of pi times the right end, 797 00:47:37,070 --> 00:47:39,550 which I'll call x_2, and here the left end, 798 00:47:39,550 --> 00:47:40,720 which I'll call x_1. 799 00:47:40,720 --> 00:47:43,860 So this is pi pi (x_2^2 - x_1^2) dy. 800 00:47:47,800 --> 00:47:50,510 Now, already at this stage, you think to yourself 801 00:47:50,510 --> 00:47:54,740 this is more complicated than the other method. 802 00:47:54,740 --> 00:47:56,500 So you've already abandoned it. 803 00:47:56,500 --> 00:47:59,790 But I'm just going to go one step further into this one 804 00:47:59,790 --> 00:48:01,860 to see what it is that's happening. 805 00:48:01,860 --> 00:48:08,950 If you try to figure out what these values x_1 and x_2 are, 806 00:48:08,950 --> 00:48:14,740 that corresponds to solving for x_1 and x_2 in terms of y. 807 00:48:14,740 --> 00:48:17,200 So that's the following equation. 808 00:48:17,200 --> 00:48:25,590 x_1 and x_2 solve the equation that-- the curve, x - 809 00:48:25,590 --> 00:48:32,080 x^3 is equal y. 810 00:48:32,080 --> 00:48:33,790 Now, look at this equation. 811 00:48:33,790 --> 00:48:42,290 That's the equation x^3-- sorry, x^3 - x + y, I guess. 812 00:48:42,290 --> 00:48:43,320 Let's see. 813 00:48:43,320 --> 00:48:46,880 Yeah, that's right, is equal to 0. 814 00:48:46,880 --> 00:48:53,370 This is a cubic equation. 815 00:48:53,370 --> 00:48:55,060 Although there is a formula for this. 816 00:48:55,060 --> 00:48:57,351 You've never been taught the formula for this equation. 817 00:48:57,351 --> 00:48:58,920 So therefore, you will never, ever 818 00:48:58,920 --> 00:49:02,560 be able to get a formula for x_2 and x_1 as a function of y. 819 00:49:02,560 --> 00:49:05,030 And you'll never be able to compute this one. 820 00:49:05,030 --> 00:49:08,030 This is more than just a dead end, it's like crash, burn, 821 00:49:08,030 --> 00:49:11,300 and, you know, self-destruct. 822 00:49:11,300 --> 00:49:14,670 So there may be such a thing, so do the other way. 823 00:49:14,670 --> 00:49:17,490 Good luck, folks.